Stability theorem for KdV-type equations

Abstract

A general KdV-type equation which covers most of the known model equations for weakly dispersive waves is investigated. First, the possible stationary localized states are discussed. When perturbed longitudinally, the perturbations can be finite and complex but should be small. In the main (second) part of the paper a necessary and sufficient stability criterion for the stationary states is derived. This is performed in two steps: A variational formulation leads to an instability region, whereas the sufficient stability criterion follows from a Liapunov functional. The stability theorem is evaluated for various models including ion-acoustic waves in plasmas and lower-hybrid cones.